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Limit comparison test:

If the positive functions f and g are continuous on $$\left [ a, \infty \right )$$ and if $$\lim_{x\rightarrow \infty }\frac{f(x)}{g(x)}=L$$ where $$0< L< \infty $$ then $$\int_{a}^{\infty }f(x) \ and \ \int_{a}^{\infty }g(x)$$ both converge or both diverge.

Now let's assume the opposite. Under the same conditions, except $$L =\left \{ 0,\infty \right \}$$

Now, can we say one of $$\int_{a}^{\infty }f(x) \ and \ \int_{a}^{\infty }g(x)$$ converges and the other diverges?

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    Consider on the convergent hand the pair $e^{-x},\; e^{-x^2}$, and on the divergent the pair $x^{-1/2},\; x^{-1/3}$. – 2017-01-17

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No you cannot. Take $f(x)=0$ and $g(x)$ to be any convergent function. Then, both integrals converge, even though $L=0$.

Or, take $f(x)=e^{-x}$ and $g(x)=\frac{1}{x^2}$. Then both integrals converge even though $L=\infty$.

The thing is that the original statement is of the type

If ($A$ and $B$), then $C$

While you want to use it to prove

If ($A$ and not $B$), then not $C$

Which is clearly not always true.

For example, "if there will be rain and storms tommorow, I will need an umbrella" is true, but "if there will be rain, but no storms tommorow, I will need an umbrella" is clearly false.

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    what if L only equals infinity? – 2017-01-17
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    @Huzo The answer is still no (see my extended answer). – 2017-01-17